Change of Basis Calculator
Convert vector coordinates from standard basis to a new basis and find the change-of-basis matrix.
Input
Vector v (in standard basis)
New Basis B = {b₁, b₂, ...}
b1
b2
[v]_B (Coordinates in Basis B)
(3.5000, -0.5000)
Linear Combination
v = 3.5000·b1 - 0.5000·b2
Change-of-Basis Matrix P
Columns are basis vectors of B
1.0001.000
1.000-1.000
P⁻¹ (Inverse)
[v]_B = P⁻¹ · v
0.5000.500
0.500-0.500
Verification: P · [v]_B = v
(3.0000, 4.0000)
Change of Basis
Formula
[v]_B = P⁻¹ · [v]_S
Where S is the standard basis
Applications
- Diagonalizing matrices
- Simplifying linear transformations
- Computer graphics transformations